Algorithms for Difference-of-Convex Programs Based on Difference-of-Moreau-Envelopes Smoothing

Abstract

In this paper, we consider minimization of a difference-of-convex (DC) function with and without linear equality constraints. We first study a smooth approximation of a generic DC function, termed difference-of-Moreau-envelopes (DME) smoothing, where both components of the DC function are replaced by their respective Moreau envelopes. The resulting smooth approximation is shown to be Lipschitz differentiable, capture stationary points, local, and global minima of the original DC function, and enjoy some growth conditions, such as level-boundedness and coercivity, for broad classes of DC functions. For a smoothed DC program without linear constraints, it is shown that the classic gradient descent method and an inexact variant converge to a stationary solution of the original DC function in the limit with a rate of [Formula: see text], where K is the number of proximal evaluations of both components. Furthermore, when the DC program is explicitly constrained in an affine subspace, we combine the smoothing technique with the augmented Lagrangian function and derive two variants of the augmented Lagrangian method (ALM), named linearly constrained DC (LCDC)-ALM and composite LCDC-ALM, targeting on different structures of the DC objective function. We show that both algorithms find an ϵ-approximate stationary solution of the original DC program in [Formula: see text] iterations. Comparing to existing methods designed for linearly constrained weakly convex minimization, the proposed ALM-based algorithms can be applied to a broader class of problems, where the objective contains a nonsmooth concave component. Finally, numerical experiments are presented to demonstrate the performance of the proposed algorithms. Funding: This work was partially supported by the NSF [Grant ECCS1751747]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoo.2022.0087 .